supplementary information for: how to optimize binding of ... · polybase consists of n b monomer...

Supplementary Information for: How to Optimize Binding of Coated

Nanoparticles: Coupling of Physical Interactions, Molecular

Organization and Chemical State

R.J. Nap and I. Szleifer#∗

#Department of Biomedical Engineering and Chemistry of Life Processes Institute,Northwestern University,

2145 Sheridan Road, Evanston, Illinois, 60208-3100, United States,

March 4, 2013

1 Theory

The system that we describe, schematically presented in Figure S1, consists of a micelle of radius R which hasend-tethered NN neutral polymers (poly ethyleneglycol) (PEG) and NB polybases (e.g., poly((dimethylamino)ethylmethacrylate) (PDMAEMA)). The degree of polymerization or length of the neutral polymer chain is nN . Thepolybase consists of nB monomer units; each unit is a basic group which can be either protonated (BH+) ordeprotonated (B). The free end of the polybase has a ligand that has specific interactions with the receptors of thelipid membrane. The micelle is positioned a distance D from a surface. The distance is measured from the centerof the sphere to the surface. The system is in a solution with fully dissociated salt, NaCl, and at a given pH, inmost cases both are physiological conditions.

We assume that the system is rotationally invariant along the axis perpendicular to the surface and passingthrough the poles of the micelle or sphere. This axial symmetry is used explicitly by employing cylindrical coordi-nates (r, ϕ, z), enabling the integration of the azimuthal angle ϕ and consequently reducing the problem from threeto two dimensions.

θ

ϕ

R D

xr

z

y

ligand

lipidcharged

receptor

Figure S1: Left: Schematic representation of a nanomicelle with functionalized head-groups. The neutral polymersare represented by blue lines and the polybases (black lines) have a ligand at the free end, (green dot) which interactswith receptors head-group (black) of lipids found in the lipid membrane. The lipid membrane consists of lipids withacidic head -groups, (phosphatidylserine). Right: Representation of the cylindrical coordinate system used. Thegray area in the right figure on the sphere represents a surface element with uniform properties.

We describe both end-tethered solid nanoparticles and micelles. For solid nanoparticles the chains are irreversiblyend-grafted to the nanoparticle; however, in the case of micelles tethered with polymers, the end-grafted polymer

∗corresponding author: email: [email protected]

1

Electronic Supplementary Material (ESI) for Biomaterials ScienceThis journal is © The Royal Society of Chemistry 2013

chains are able to diffuse on the surface, i.e., have translational entropy1, since they are part of the surfactant headgroup. The mobility of the chains implies that we have to allow for an inhomogeneous distribution of the polymerson the micelle. Therefore it is convenient to introduce the local surface of coverage: σi(θ) = Ni(θ)/A(θ), whichcorresponds to the number of polymers of type i tethered to a surface sector θ divided by the area of the surfacesector. See figure S1. In case of an homogenous polymer distribution the local surface coverage, σi(θ), equals theglobal surface coverage, σi = Ni/Asurf. Here Asurf is the total surface area of the sphere. Otherwise the total surfacecoverage is given by the integral σi =

∫dA(θ)Ni(θ)/Asurf.

The surface consists of a mixture of neutral and charged lipids. The lipids on the membrane are characterizedby the areal surface density of their components, which are denoted by σN (r) and σC(r), respectively referringto the neutral and negatively charged lipid. The later molecule can be found either in its protonated (AH) ordeprotonated (A−) state; σC(r) = σA−(r) + σAH(r). Each, neutral and anionic lipids have an area per lipid ofa0 = 0.65 nm2. Furthermore, the lipid layer may contain overexpressed receptors. We only consider the case inwhich the receptors are in large excess and therefore we do not need to consider their density explicitly.

The total free energy describing the system has the following contributions

F = −TSconf − TSmix,poly − TSmix + Erep + Eelec + Fchem +Gbind + Fsurf, (1)

where T is the temperature. In the following we describe each of the terms of the free energy in terms of theprobability functions and local varying densities. The first term in the free energy represents the conformationalentropy of the polymer chains and is given by

−Sconf(D)

kB=

∫dθ NN (θ)

∑αN

P (αN , θ,D) lnP (αN , θ,D)

+

∫dθ NB(θ)

∑αB

P (αB , θ,D) lnP (αB , θ,D) (2)

The first term denotes the conformation entropy of the neutral chains whereas the second term represent theconformation entropy of the basic polymer chains.

Here, P (αi, θ,D) is the probability of finding a type i polymer chain tethered to the surface element θ andmicelle lipid membrane distance D in a conformational state αi. Ni(θ)dθ is the number of polymer chains of type itethered to the surface sector located in the interval (θ, θ+ dθ). The micelle and the lipid membrane are a distanceD apart. See fig. S1.

The second term in the free energy accounts for the translational entropy of the mobile chains relative to theirentropy for a homogeneous surface distribution of the chains; σi = Ni/Asurf .

−Smix,poly(D)

kB=

∫dθ NN (θ)(lnσN (θ)/σN − 1) +

∫dθ NB(θ)(lnσB(θ)/σB − 1). (3)

In case of immobile tethers the local surface coverage is fixed and this translational entropy contribution to the freeenergy is constant since σB(θ) = σB and σN (θ) = σN .

The third term in the free energy denotes the mixing (translational) entropy of all other species in the system;i.e., water, cations, anions, protons, and hydroxyl ions.

−Smix(D)

kB=

∫∫dr dzG(r, z)ρw(r, z)(ln ρw(r, z)vw − 1)

+

∫∫dr dzG(r, z)ρNa+(r, z)(ln ρNa+(r, z)vw − 1)

+

∫∫dr dzG(r, z)ρCl−(r, z)(ln ρCl+(r, z)vw − 1)

+

∫∫dr dzG(r, z)ρH+(r, z)(ln ρH+(r, z)vw − 1 + βµ

H+)

+

∫∫dr dzG(r, z)ρOH−(r, z)(ln ρOH−(r, z)vw − 1 + βµ

OH−). (4)

Here ρj(r, z) is the number density of molecular species j at position (r, z) and G(r, z)drdz corresponds to thevolume element that in cylindrical coordinates is equal to 2πrdrdz. The terms corresponding to the protons andhydroxyl atoms include also the standard chemical potential (µ

i ) of those species as these terms are necessary forthe proper consideration of the chemical equilibrium, see below.

2


The fourth term, Erep, of the free energy describes the intermolecular excluded volume interactions, which areaccounted for by assuming that the system is incompressible at every position. Namely, that the sum of the volumefraction of polymers, solvent, and ionic species add up to one at every postition;

〈φN (r, z)〉+ 〈φB(r, z)〉+ ρw(r, z)vw + ρNa+(r, z)vNa+ + ρCl−(r, z)vCl−

+ ρH+(r, z)vH+ + ρOH−(r, z)vOH− + φM (r, z) = 1. (5)

These packing constraints are enforced in the minimization of the free energy through the introduction of theLagrange multipliers π(r, z), see below. Here 〈φN (r, z)〉 and 〈φB(r, z)〉 correspond to the volume fraction of theneutral polymer and the polybase. These volume fractions are given by :

〈φi(r, z)〉 =

∫dθ

Ni(θ)

G(r, z)

∑αi

P (αi, θ,D)ni(αi, θ; r, z)vpi . (6)

Here ni(αi, θ; r, z)drdz is the number of segments of polymer type i in the volume element G(r, z)drdz whenits conformation αi is tethered to surface area element A(θ)dθ. The area element A(θ) is given by A(θ)dθ =2π sin(θ)R2dθ. vpi denotes the volume of one polymer segment of type i. In the packing constraint Eq. 5, φM (r, z)denotes the volume fraction of the micellar core, which is introduced to allow for solvent and ions to penetrate themicellar core. We assume φM (r, z) = φM = constant for those r, z coordinates inside the micellar core. Throughoutthe calculation we set φM = 0.9. Note that the results are insensitive to the particular choice of φM .

The hydrophobicity of the polymers, i.e., the solvent quality, can be described by an effective attractive Vander Waals interaction2

EVdW =1

2

∫d~r

∫d~r′χp(|~r − ~r′|)〈φp(r, z)〉〈φp(r′, z′)〉. (7)

We limit ourself to good-solvent conditions (χp = 0) because the polymers we model, PEG and PDMAEMA, arewater soluble, i.e., water acts as a good solvent for them.

The fifth term in equation 1 describes the electrostatic energy contribution to the free energy3

Eelect(D) = β

∫∫dr dz G(r, z)

(ρq(r, z)ψ(r, z)− 1

2ε(r, z)(∇ψ(r, z))2

)+ β

∫dr A(r)σq(r)ψ(r, 0), (8)

here ψ(r, z) corresponds to the electrostatic potential and ρq(r, z) to the charge density. The charge density is thesum of the charge arising form the protonated polybase units and the free ionic species and is given by

ρq(r, z) = f(r, z)〈ρB(r, z)〉e+ ρNa+(r, z)e− ρCl−(r, z)e+ ρH+(r, z)e− ρOH−(r, z)e, (9)

where e is the unit of charge, 〈ρB(r, z)〉 equals the density of polybase monomers, and f(r, z) corresponds to thefraction of polybase segments that are charged at r, z. The surface charge density σq(r) refers to the amount ofcharged lipid molecules found on the model cell membrane at r, to be discussed in more detail below.

The free energy associated with the (de)protonation of the base group of the polymer (B + H+ � BH+) isrepresented by4,5

βFchem(D) = β

∫∫dr dz〈ρB(r, z)〉

(f(r, z)(ln f(r, z) + βµ

BH+)

+(1− f(r, z))(ln(1− f(r, z)) + βµB)) (10)

The first and third term describe the entropy of mixing between the protonated and deprotonated states at each r, z.The second and fourth term represent the standard chemical potential of forming a protonated and a deprotonatedstate respectively.

The seventh contribution, Gbind, to the free energy describes the free energy change associated with the bindingof a ligand attached to the end-group of the polybase chain to a receptor located on the lipid membrane. It isassumed that the number of receptor sites on the membrane are in excess corresponding to conditions for which asurface has overexpressed receptors. Then, the free energy contribution of the ligand-receptor binding is given bythe number of end-groups (neB(αB , θ; r, 0)) or ligand-receptor pairs at the membrane times the free energy to formone ligand-receptor bond, ∆Gbind.

βGbind(D) = β

∫∫dr dz G(r, z)

∫dθ Ni(θ)

∑α=αB

P (αB , θ,D)neB(α, θ; r, z)∆Gbindδ(z). (11)

3


The free binding energy is expressed in units of kBT . It is possible to take into consideration with the theory caseswhere the amount of receptors are not in excess and include within the theoretical framework both the amount ofreceptors on the surface as well as the fraction of ligand that bind.6,7

The last term of the free energy functional labeled Fsurf represents the free energy contribution arising from thesurface lipids8–11.

βFsurf(D) =

∫dr A(r)σN (r)(ln(σN (r)/σl)− 1) + σC(r)(ln(σC(r)/σl)− 1)

+

∫dr A(r)σC(r)

[x(r)(lnx(r) + βµ

A−) + (1− x(r))(ln(1− x(r)) + βµAH)

]. (12)

The first two terms in Fsurf represent the (translational) mixing entropy of the mixture of neutral and chargedlipid molecules present on the membrane. Here σN (r) and σC(r) are the local areal surface density of the headgroups of the neutral and anionic lipids. The lipid membrane is covered with neutral and charged lipids; hencethe areal density of both lipids at every position on the membrane add up to the total areal lipid density; σl =σC(r) + σN (r). These constraints are enforced through the introduction of Lagrange multipliers πs(r) in the freeenergy minimization. The neutral and anionic lipid have the same area per lipid a0 = 0.65nm2. Consequentlyσl = 1/a0 = 1.54nm2. The anionic lipid is considered to be a weak acid (AH � A− + H+), thus it is eitherprotonated (AH) of deprotonated (A−) depending on its local environment. The energy and entropic contributionarising from this acid-base chemical equilibruim are accounted for by equation 12. Here x(r) is the fraction ofacidic lipids that are charged (deprotonated) at r: x(r) = σA−(r)/(σA−(r) + σAH(r)) = σA−(r)/σC(r). Followingthis definition the total surface charge density at position r becomes σq(r) = −ex(r)σC(r). The standard chemicalpotential µ

AH and µA+ in Eq. 12 denote the free energy of the protonated and deprotonated state respectively and

the remaining terms represent the entropy of mixing associated with the protonated and deprotonated state.The above form of Fsurf corresponds to a lipid membrane in the liquid-disordered phase, i.e., the lipids are

mobile, and the charged lipids are charge regulating (weak acids). Expressions for lipids membranes in a gel phasewhere the lipids can not move or where the anionic lipids are not charge regulating can be obtained from the aboveexpression by appropriately considering constant lipid areal density or a constant charged lipid fraction. Fourdifferent scenarios have been considered, namely the lipid membrane is either in the liquid-disorder or gel phase(mobile or immobile) and the charged lipids are either charge regulating or not.

To establish the thermodynamic equilibrium state, the free energy is minimized with respect to the Pi(α, θ),ρi(r, z), Ni(θ), f(r, z), x(r), σN (r), and σC(r), and varied with respect to the electrostatic potential ψ(r, z) underthe constraints of incompressibility, mass balance of the polymer chains, and the fact that the system is in contactwith a bath of cations, anions, protons, hydroxyl atoms, and lipids. The proper thermodynamic potential becomes:

βW = βF + β

∫∫dr dz G(r, z)π(r, z) (〈φN (r, z)〉+ 〈φB(r, z)〉+ ρw(r, z)vw + ρNa+(r, z)vNa+

+ρCl−(r, z)vCl− + ρH+(r, z)vH+ + ρOH+(r, z)vOH− + φM (r, z)− 1)

+β

∫dr A(r)πs(r) (σN (r) + σC(r)− σl)

−βµN∫

dθ (NN (θ)−NN )− βµB∫

dθ (NB(θ)−NB)

−∑

i=Na+,Cl−

βµi

∫∫dr dz G(r, z)ρi(r, z)− µC

∫dr A(r)σC(r). (13)

The number of independent thermodynamic components is reduced by three, because the system is incompressible,charge neutral, and the water molecules are in chemical equilibrium with the protons and hydroxyls atoms. Dueto these constraints, the chemical potentials are exchange chemical potentials. Equivalently, the area constraintreduces the number of independent surface components. Therefore it is unnecessary to explicitly introduce theexchange chemical potential for both the neutral and anionic lipid. For a more extended discussion on the subjectof exchange chemical potentials see references2 and5. The total number of polybase and neutral polymer chains arefixed at a value of NB and NN respectively; i.e.,

∫Ni(θ)dθ = Ni. The Lagrange multipliers µB and µN introduced

to impose these (mass balance) constraints for the polymers can be identified as the chemical potentials of thepolybase and the neutral polymer respectively.

Minimization yields for the probability distribution function of the neutral (PEG) molecule

P (αN , θ,D) =1

qN (θ,D)exp

[−β∫∫

nN (αN , θ; r, z)π(r, z)vpNdrdz

], (14)

4


here qN (θ,D) is a normalization factor ensuring that the probability distribution function is properly normalized:∑αN

P (αN , θ,D) = 1. The term in the exponential results from the repulsive interactions that a neutral polymerin conformation αN experiences.

Minimization yields for the probability distribution function of a polybase molecule the following equation

P (αB , θ,D) =1

qB(θ,D)exp

[−β∫∫

nB(αB , θ; r, z)π(r, z)vpBdrdz − β∫neB(αB , θ; r, 0)∆Gbinddr

]× exp

[−∫∫

nB(αB , θ; r, z) (βψ(r, z)e+ ln(f(r, z))) drdz

]. (15)

The first term in the exponential is identical in meaning to the one appearing in the pdf of the neutral polymer. Thesecond term is an attractive contribution arising from the binding of the functional end-group of the polymer chainwhen in contact with the surface. This term only contributes to P (αB , θ,D) when the end-group of conformationαB for a given anchor location θ reaches the membrane. The remaining two terms represent contributions arisingfrom the electrostatic interaction and the chemical equilibrium. The probability distribution functions, like othervariables such as the densities, also depend on the distance D between the micelle and the lipid membrane.

Minimization of the free energy with respect to the local number of tethered polymers or equivalently the localsurface coverage gives

βµi(D) = − ln(qi(θ,D)) + ln(σi(θ)/σi). (16)

The surface coverage of polymers of type i is

σi(θ) = σieβµi(D)qi(θ,D) with eβµi(D) =

Asurf∫A(θ)qi(θ,D)dθ

. (17)

The second equation is obtained by integration over the surface of the sphere and using the relations∫A(θ)σi(θ)dθ =

Ni and∫A(θ)dθ = Asurf. The above equations guarantee that the chemical potential, µi, of the polymer chains

of type i is the same for every position along the micelle’s surface, as required by thermodynamic equilibrium.The equations also demonstrate the non-local coupling that exists between the polymer chains tethered at differentgrafting positions and show that the chemical potential will change with the separation between micelle and lipidmembrane. As the distance is reduced, the polymer layer get confined. The confinement results in a change of theprobability distribution function of the chains that causes a change of the chemical potential of the polymers.

By minimizing the free energy with respect of degree of dissociation of the polybase we obtain

f(r, z)

1− f(r, z)= K

b

φH+(r, z)

φw(r, z). (18)

Here Kb = exp (−β∆G

b ) with ∆Gb = µ

BH+ − µB − µ

H+ which is equal to the standard free energy of thechemical reaction (B+H+ � BH+). K

b is the equilibruim constant of the chemical reaction and is related to theequilibrium constant in bulk solution Kb = [BH+][OH−]/[B] = CKwK

b . Here [] denotes the molar concentration

and Kw is the water equilibrium constant. The constant C is needed for consistency of units as Kb has the unitsof molarity whereas K

b is dimensionless (C = 1/(NAvw)).Variation of the free energy functional with respect to the electrostatic potential ψ(r, z) leads to the Poisson

equation and its boundary conditions:3,5

∇(ε(r, z)∇ψ(r, z)) = −〈ρq(r, z)〉 ∧ ε∇ψ · nS |∂V = −σq(S). (19)

Here ∇ corresponds to the gradient or del operator in cylindrical coordinates. The second term represent theboundary conditions with the lipid surface, the micellar core, as well as the boundary conditions with the bulksolution. The explicit electrostatic boundary condition for the lipid surface (z = 0) reads:

−εwε0∂ψ(r, z)

∂z

∣∣∣∣z=0

= σq(r). (20)

For z → ∞ we can set limz→∞ ψ(r, z) = 0. For general z we need to use ∂∂rψ(r, z) = 0 as only when there is no

charge on the surface (σq(r) = 0) does limr→∞ ψ(r, z) = 0.Minimization of the free energy with respect of solvent density yields

φw(r, z) = ρw(r, z)vw = exp (−βπ(r, z)vw) . (21)

5


The physical meaning of the Lagrange multipliers can be understood from the last expression. They correspond tothe position dependent osmotic pressures. One can also view them as the average interaction or potential due tothe excluded volume interactions. A more detailed discussion on the physical significance of these quantities can befound elsewhere2.

The expressions for the densities of the cations and anions are

ρNa+(r, z)vw = exp (βµNa+ − βπ(r, z)vNa+ − βψ(r, z)e) , (22)

ρCl−(r, z)vw = exp (βµCl− − βπ(r, z)vCl− + βψ(r, z)e) , (23)

while the density of the proton and hydroxyl ions are given by

ρH+(r, z)vw = exp(βµ

H+ − βπ(r, z)vH+ − βψ(r, z)e), (24)

ρOH−(r, z)vw = exp(βµ

OH− − βπ(r, z)vOH− + βψ(r, z)e). (25)

In deriving the above equations we assumed that the system is in contact with a bath of ions. Therefore, as aconsequence of thermodynamic equilibrium, the chemical potentials of the ions are constant at every position andtheir values are determined by the bulk conditions.

By minimizing the free energy with respect of the degree of dissociation of the acidic lipid x(r) we get

x(r)

1− x(r)= K

a

φw(r, 0)

φH+(r, 0)= K

a exp(βµH+) exp(βψ(r, 0)e). (26)

This expression is similar to the equation for the degree of dissociation of the polybase. Here Ka = exp (−β∆G

a )with ∆G

b = µA− + µ

H+ − µAH which is equal to the standard free energy of the chemical reaction. K

a is theequilibrium constant of the acid-base chemical reaction, AH � A−+H+, and is related to the equilibrium constantin bulk solution Ka = [A−][H+]/[AH] = CK

a . Observe that the second part of the chemical equilibrium equationis presented to stress the similarity with the chemical equilibrium equation of the polybase. In practice the thirdequation is used, which depends only on the surface electrostatic potential and µ

H+ which can be related to thebulk concentration of the protons; i.e., pH = − log[H+].

For the density of the neutral and anionic lipids we get

σN (r)/σl = exp(−βπs(r)), (27)

σC(r)/σl = exp(−βπs(r)− βψ(r, 0)(−e) + lnx(r)− β(µA− + µC)). (28)

Compare the above equations with the equations obtained for the ions and the probability distribution functions.They are similar in content and meaning. The value of the chemical potentials can be determined from the bulkcomposition of the lipid membrane. It turns out to be convenient to introduce the following variable: fl(r) =σC(r)/(σN (r) + σC(r) = σC(r)/σl, which is equal to the fraction of acidic lipids in the membrane at position r.Consequently σC(r) = fl(r)σl , σN (r) = (1− fl(r))σl and fl(r) as a function of x(r) becomes

fl(r)

1− fl(r)=e−β(µ

AH−µC)

1− x(r)(29)

The above equation can be readily solved once the bulk composition f bulkl and surface potential ψ(r, 0) are known.Having found the equations describing the thermodynamical equilibrium the minimal free energy can be ob-

tained. The effective interaction between the micelle and the surface is given by

β∆W (D) = βW (D)− βW (∞)

= β(µN (D)− µN (∞))NN + β(µB(D)− µB(∞))NB

−1

2β

∫∫drdzG(r, z) (〈ρq(r, z)〉ψ(r, z)− 〈ρq,∞(r, z)〉ψ∞(r, z))

−β∫∫

drdzG(r, z) (π(r, z)(1− φM (r, z))− π∞(r, z)(1− φM,∞(r, z)))

−∑

i=w,Na+,Cl−,H+,OH−

∫∫drdzG(r, z)(ρi(r, z)− ρi,∞(r, z))

−β∫

drA(r)(πs(r)− πs,∞)− 1

2β

∫A(r) (σq(r)ψ(r, 0)− σq,∞ψ∞(r, 0)) , (30)

6


which corresponds to the difference between the free energy when the micelle and the surface are a distance D apartand when they are infinitely far apart. Equation 30 is obtained by substituting the equations for the densities ofthe ions, solvent, lipids, the pdfs and local surface coverage of the polymers, the electrostatic potential, and thedegree of dissociation of both polybase and acidic lipid into the free energy expression (Eq. 13).

The unknowns in Eqs. 21-29 and 14-18 are the lateral pressures and electrostatic potential.5,12 Application ofthe theory requires the determination of these lateral pressures and electrostatic potential. The unknown lateralpressure and electrostatic potential can be obtained by substituting the volume fractions and areal densities ofthe lipids into the incompressibility constraint, Eq. (5), the Poisson Eq. (19) and the Eq. (29 and 26), thelater two equations determine the composition of the lipid membrane. The resulting set of coupled non-linearintegrodifferential equations can be discretized and solved numerically. Technical aspects of how to apply andnumerically solve the theory are outlined in the next section. The input necessary to solve the equations are thesurface coverage of the neutral polymers and polybases, σN and σB , the binding energy ∆Gbind and the radius Rof the nanomicelle, the distance D between nanomicelle and surface, and the sets of polymer conformations for theneutral polymer and polybase, αN and αB , the pKb of the polybase, the pKa of the acidic lipid, the bulk fractionof neutral and acidic lipids, and the salt concentration and pH of the bulk solution.

2 Numerical methodology

A numerical solution for the position dependent lateral pressure π(r, z) and electrostatic potential ψ(r, z) isobtained by discretization of the packing constraints, Eq. (5), and the Poisson Eq. (19). The equations arediscretized by dividing the rz-plane into a grid of squares of length δ. Functions are assumed to be constant withina grid cell, hence integrations can be replaced by summations. The integral of a general position dependent functionf(r, z) then becomes:∫ ∫

drdz G(r, z)f(r, z) =∑i,j

∫ iδ

(i−1)δdr

∫ jδ

(j−1)δdz G(r, z)f(r, z) ≈

∑i,j

f(i, j)∆G(i, j), (31)

with

∆G(i, j) =

∫ iδ

(i−1)δdr

∫ jδ

(j−1)δdz G(r, z). (32)

Here f(i, j) denotes the value which function f(r, z) attains within the region located between (i−1)δ ≤ r < iδ and(j − 1)δ ≤ z < jδ, and G(r, z)drdz = 2πrdrdz corresponds to a volume element. The geometric factor ∆G(i, j) isthe finite volume of the discrete cell. The calculations can be carried out outside the micelle (sphere with excludedvolume interactions) and the derived expressions are strictly valid in that region of space. For discretization of thePoisson equation, to be discussed below, it turns out to be convenient to consider also the inside of the micellarcore. This requires a small modification of the discretization of the grid cells. Namely, grid cells (i, j) whose regioncoincide with the boundary of the micellar core are split in two cells; one that is strictly on the outside and theother that is strictly on the inside. For those grid cells the integration boundary needs to be modified to accountfor the presence of the boundary of the micellar core. For grid cells (i, j) strictly located outside/inside the sphere:∆G(i, j) = π(2j − 1)δ3.

Integration of the area of the sphere are discretized as follows,∫dθA(θ)g(θ) ≈

∑k

g(k)∆A(k), (33)

with ∆A(k) =∫ θkθk−1

dθ A(θ). The integration boundaries are determined by the discretization of the grid.

The packing constraint, Eq. (5), in discrete form for grid cell (i, j) read

〈φN (i, j)〉+ 〈φB(i, j)〉+ φw(i, j) + φNa+(i, j) + φCl−(i, j) + φH+(i, j) + φOH−(i, j) + φM (i, j) = 1. (34)

7


Eqs. (21), (6), (14), (15), and (17) in discrete form become

φw(i, j) = exp(−βπ(i, j)vw), (35)

〈φγ(i, j)〉 =∑k

∆A(k)σγ(k)

∆G(i, j)

∑αγ

P (αγ , k,D)n(αγ , k; i, j)vp (36)

n(αγ , k; i, j) ≡∫ iδ

(i−1)δdr

∫ jδ

(j−1)δdz n(αγ , θ(k); r, z), (37)

PN (αN , k), D =1

qN (k,D)

∏i,j

exp[−βπ(i, j)n(αN , k; i, j)vpN ], (38)

PB(αB , k,D) =1

qB(k,D)

∏i,j

exp[−βπ(i, j)n(αB , k; i, j)vpB − β∆Gbindne(αB , k; i, 1)]

× exp [−n(αB , k; i, j) (eβψ(i, j) + ln(f(i, j))] , (39)

σγ(k) = σγeβµγ qγ(k,D), (40)

eβµγ(D) = Asurf/∑k

qγ(k,D)∆A(k). (41)

The volume fraction of the counter, co-ions, protons, and hydroxyl ions, ( Eqs.(22), (23), (24), and (25) ), indiscretized space are:

φNa+(i, j) = φNa+,bulk exp(−β(π(i, j)− πbulk)vNa+ − eβψ(i, j)), (42)

φCl−(i, j) = φCl−,bulk exp(−β(π(i, j)− πbulk)vCl− + eβψ(i, j)), (43)

φH+(i, j) = φH+,bulk exp(−β(π(i, j)− πbulk)vH+ − eβψ(i, j)), (44)

φOH−(i, j) = φOH−,bulk exp(−β(π(i, j)− πbulk)vOH− + eβψ(i, j)). (45)

The above volume fractions depend on the lateral pressure, electrostatic potential and on the bulk volume fractions.The chemical potentials of the ions, protons, and hydroxyl ions are related to their bulk volume fractions.5 Thesebulk values are input to the theory.

The discrete form of the degree of dissociation of the acidic lipid is given by

x(i)

1− x(i)= K

a exp(βµH+) exp(βψ(i, z = 0)e), (46)

and the fraction of acidic lipids in the lipid membrane is given by

fl(i)

1− fl(i)=e−β(µ

AH−µC)

1− x(i), (47)

here ψ(i, z = 0) is the surface electrostatic potential of the lipid membrane surface at (ri, z) = ((i− 1/2)δ, 0).The discretized Poisson equation, in cylindrical coordinates, is(1 +

δ

2ri

)ψ(i+ 1, j)− 4ψ(i, j) +

(1− δ

2ri

)ψ(i− 1, j)

+ ψ(i, j + 1) + ψ(i, j − 1) = −εwε0δ2ρq(i, j), (48)

here ri = (i−1/2)δ denotes the middle of the grid cell (i, j) in the radial direction. The above equation only appliesfor regular grid cells, i.e., those grid cells not at the boundary with the micellar core. For the split “non-regular”grid cells close to the micellar core/NP and the boundary condition of the interface of the micellar core we needto employ a generalized 5-point “stencil” method. Details concerning the constructions of those “stencils” canbe found in Ref.13. The above scheme is computationally complex and difficult to implement; moreover, overallcharge neutrality is not guaranteed. To address this problem and reduce the complexity of the algorithm we appliedan approximation of the micellar electrostatic boundary condition. The electrostatic discontinuity of the micellarboundary was “ignored” and we combined the charge density of the “split” non-regular grid cells into one regularcell. Conceptually this means that we “smear” out the charges near the interface of the micellar core. We found thatthe difference in free energy computed with both approaches is less then 1% at maximum, clearly demonstrating

8


the relative unimportance of the electrostatic boundary condition near the micellar core. On the other hand, theelectrostatic boundary conditions near the lipid membrane are of prime importance as the result presented in themain paper demonstrate. The discretized electrostatic boundary conditions for the surface (z = 0) read:

ψ(i, 1)− ψ(i, z = 0) = −δεwε0σq(i). (49)

Here ψ(i, z = 0) is the surface electrostatic potential of the lipid membrane surface.Substituting Eqs. (37), through (45) into the constraint equation (34) and the Poisson Eq. (48) and substituting

of Eqs (47) and (46) into the electrostatic boundary condition of the lipid surface (Eq. 49) results in a set of couplednonlinear equations which can be solved by standard numerical methods.14

3 Chain Model

We use the three-state rotational isomeric state (RIS) model of Flory15 in order to generate a representativesample set of chain conformations. These conformations are generated by a simple sampling procedure that takesinto account the self-avoidance of the polymer chain. Through appropriate translational and rotational adjustments,the generated chain conformations are end-tethered to the center of a surface element. In doing so, both the surfacesof the sphere and the external plane are assumed to be non-penetrable by the chain segments. The segment ormonomer length of the polymers and the volume of one polymer segment are l = 0.35nm, vpN = 0.065nm3 for theneutral PEG polymer and l = 0.35nm, vpB = 0.113nm3 for the polybase polymer respectively. The volume of thewater molecules is vw = 0.03nm3. The proton and hydroxyl atom have a similar volume; vH+ = vOH− = vw. Theanion and cation have a volume of vCl− = vNa+ = 0.0335nm3. We used a discretization length δ = 0.5nm.

One set of chain conformations is generated for both the neutral and polybase and is used for all calculationsreported in this paper. The number of conformations per chain type at each grafting position is 1000000. For amicellar core radius of R = 2.5nm this amount to 2.8 ∗ 107 chain conformations. Note that this large number ofconformations requires the parallelization of the code.

4 Results

2 4 6 8D -R (nm)

-2

-1

0

1

2

∆W

(kBT

)/n

xPS

=0.065 ∆Gbind

= 0 kBT

xPS

=0 ∆Gbind

= -5kBT

xPS

=0.065 ∆Gbind

= -5kBT

2 4 6 8D -R (nm)

2 4 6 8D -R (nm)

0 2 4 6 80

10

20

30

pH=7.0 pH=7.5 pH=8.0

Figure S2: Free energy versus separation distance between charged polymer grafted planar surface andmembrane at pH = 7.0, 7.5, and 8.0. The condition used for the planar surface are identical to the one used forthe coated micelle. The grafted surface has a surface coverage for both PEG and polybase of σ = 0.20 nm−2. BothPEG and polybase have n = 20 segments. The polybase has a pKb = 6.5. The intrinsic pKa of the PS lipids is 3.6.The salt concentration is cs = 0.10M . Because of the planar geometry the free energy presented as the free energyper polymer chain. The inset in the middle panel shows the same free energy versus distance but on a zoomed outscale, showing the larger (conformation) repulsion (and divergence at short distances) of the interaction betweenthe two planar surfaces. Observe also that the effect of pH is greatly diminished: with decreasing pH the free energycurve only slightly shift to shorter distances.

9


-15 -10 -5 0 5 10 15

x (nm)

-15

-10

-5

0

5

10

15

y (

nm

)

0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

Figure S3: Contour map of the distribution of the fraction of charged PS head group of the membranelipids: (σC(r)/σ0). The center of the micelle is located at (x, y) = (0, 0) and a distance D − R = 2.0nm abovethe the membrane surface. The core of the micelle has a radius of R = 2.5 nm and the surface coverage of bothPEG and polybase is σ = 0.20 nm−2. Both PEG and polybase have n = 20 segments. The polybase has a pKb of6.5. The salt concentration is cs = 0.10M and pH = 7.5. The intrinsic pKa of the PS lipids is pKa of 3.6. Thelipids under the adhering micelle are in equilibrium with a bulk lipid membrane that possesses 6.5% of lipids withan acidic headgroup (xPS = σC/σ0 = 0.065).

0 2 4 6D-R (nm)

-30

-20

-10

0

10

20

30

∆W

(kBT

)

xPS

=0.065 ∆Gbind

= 0 kBT

xPS

=0 ∆Gbind

= -5 kBT

xPS

=0.065 ∆Gbind

= -5 kBT

0 2 4 6D-R (nm)

0 2 4 6D-R (nm)

pH=7.0 pH=7.5 pH=8.0

BA C

Figure S4: Free energy versus separation distance between charged polymer grafted solid nanoparticleand membrane at pH = 7.0, 7.5, and 8.0. The solid nanoparticle has a radius of R = 2.5 nm and the surfacecoverage of both PEG and polybase is σ = 0.20 nm−2. Both PEG and polybase have n = 20 segments. Thepolybase has a pKb = 6.5. The intrinsic pKa of the PS lipids is 3.6. The salt concentration is cs = 0.10M .

10


2 4 6 8 10 12pH

10-50

10-40

10-30

10-20

10-10

Kd

is

xPS

= 0.000

xPS

= 0.065

xPS

= 0.130

xPS

= 0.195

Figure S5: Dissociation constant versus pH for a charged micelle interacting with a liquid-like lipidmembrane with different amounts of chargeable PS head-groups. The amount of charge of the PS head-group isdetermined by an acid-base equilibrium with an intrinsic pKa of 3.6. All remaining parameters and conditions areidentical to the ones presented in Fig S3.

0 0.05 0.1 0.15 0.2 0.25x

PS

10-40

10-30

10-20

10-10

100

Kd

is

charge regulation, PS fixedcharge regulation, PS mobile

Figure S6: Dissociation constant versus amount of chargeable PS head-groups for a charged micelleversus amount interacting with for two different ’types’ of lipid membranes. The black line correspond toa gel like membrane (charges can not move), whereas the red line correspond to a liquid-like membrane (charges aremobile). In both cases the amount of charge is determined by an acid-base equilibrium. All remaining parametersand conditions are identical to the ones presented in Fig S3.

11


References

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[13] A. R. Mitchell and D. F. Griffiths, The Finite Difference Method in Partial Differential Equations, Wiley,Chichester, 1980.

[14] A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D. E. Shumaker and C. S. Woodward, ACMTrans Math Software, 2005, 31, 363–396.

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